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Offline algorithms for dynamic minimum spanning tree problems
We describe an efficient algorithm for maintaining a minimum spanning tree (MST) in a graph subject to a sequence of edge weight modifications. The sequence of minimum spanning trees is computed offline, after the sequence of modifications is known. The algorithm performs (log n) work per modification, where n is the number of vertices in the graph. We use our techniques to solve the offline geometric MST problem for a planar point set subject to insertions and deletions; our algorithm for this problem performs O(log^2 n) work per modification. No previous dynamic geometric MST algorithm was known
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Fully dynamic maintenance of euclidean minimum spanning trees
We maintain the minimum spanning tree of a point set in the plane, subject to point insertions and deletions, in time O(n^5/6 log1^2/2 n) per update operation. No nontrivial dynamic geometric minimum spanning tree algorithm was previously known. We reduce the problem to maintaining bichromatic closest pairs, which we also solve in the same time bounds. Our algorithm uses a novel construction, the ordered nearest neighbors of a sequence of points. Any point set or bichromatic point set can be ordered so that this graph is a simple path
Faster Worst Case Deterministic Dynamic Connectivity
We present a deterministic dynamic connectivity data structure for undirected
graphs with worst case update time and constant query time. This improves on the previous best
deterministic worst case algorithm of Frederickson (STOC 1983) and Eppstein
Galil, Italiano, and Nissenzweig (J. ACM 1997), which had update time
. All other algorithms for dynamic connectivity are either
randomized (Monte Carlo) or have only amortized performance guarantees
Optimal decremental connectivity in planar graphs
We show an algorithm for dynamic maintenance of connectivity information in
an undirected planar graph subject to edge deletions. Our algorithm may answer
connectivity queries of the form `Are vertices and connected with a
path?' in constant time. The queries can be intermixed with any sequence of
edge deletions, and the algorithm handles all updates in time. This
results improves over previously known time algorithm
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